Prove identity then evaluate integral

A question is this type if and only if it has multiple parts where the first part proves a trigonometric identity, and subsequent parts use that identity to evaluate definite integrals.

4 questions · Standard +0.3

1.05l Double angle formulae: and compound angle formulae
Sort by: Default | Easiest first | Hardest first
CAIE P3 2010 June Q4
6 marks Standard +0.3
4
  1. Using the expansions of \(\cos ( 3 x - x )\) and \(\cos ( 3 x + x )\), prove that $$\frac { 1 } { 2 } ( \cos 2 x - \cos 4 x ) \equiv \sin 3 x \sin x$$
  2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin 3 x \sin x \mathrm {~d} x = \frac { 1 } { 8 } \sqrt { } 3$$
CAIE P3 2019 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-16_524_689_260_726} The diagram shows the curve \(y = \sin 3 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) and its minimum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. By expanding \(\sin ( 3 x + x )\) and \(\sin ( 3 x - x )\) show that $$\sin 3 x \cos x = \frac { 1 } { 2 } ( \sin 4 x + \sin 2 x ) .$$
  2. Using the result of part (i) and showing all necessary working, find the exact area of the region \(R\).
  3. Using the result of part (i), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\cos 2 x\) and hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2021 November Q6
6 marks Standard +0.3
6
  1. Using the expansions of \(\sin ( 3 x + 2 x )\) and \(\sin ( 3 x - 2 x )\), show that $$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )\).
OCR C4 Q9
8 marks Standard +0.3
  1. Express \(\cos\theta + \sqrt{3}\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(\alpha\) is acute, expressing \(\alpha\) in terms of \(\pi\). [4]
  2. Write down the derivative of \(\tan\theta\). Hence show that \(\int_0^{\frac{\pi}{3}} \frac{1}{(\cos\theta + \sqrt{3}\sin\theta)^2} \, d\theta = \frac{\sqrt{3}}{4}\). [4]